Integer, Intent (In)	::	key, limlst, lw, liw
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	lst, ierlst(limlst), iw(liw)
Real (Kind=nag_wp), External	::	g
Real (Kind=nag_wp), Intent (In)	::	a, omega, epsabs
Real (Kind=nag_wp), Intent (Out)	::	result, abserr, erlst(limlst), rslst(limlst), w(lw)

C Header Interface

#include <nag.h>

void

d01asf_ (
double (NAG_CALL *g)(const double *x),
const double *a, const double *omega, const Integer *key, const double *epsabs, double *result, double *abserr, const Integer *limlst, Integer *lst, double erlst[], double rslst[], Integer ierlst[], double w[], const Integer *lw, Integer iw[], const Integer *liw, Integer *ifail)

The routine may be called by the names d01asf or nagf_quad_dim1_inf_wtrig.

3 Description

d01asf is based on the QUADPACK routine QAWFE (see Piessens et al. (1983)). It is an adaptive routine, designed to integrate a function of the form

g (x) w (x)

over a semi-infinite interval, where

w (x)

is either

\sin (ω x)

\cos (ω x)

Over successive intervals

C_{k} = [a + (k - 1) c, a + k c], k = 1, 2, \dots, lst

integration is performed by the same algorithm as is used by d01anf. The intervals

C_{k}

are of constant length

c = {2 [| ω |] + 1} π / | ω |, ω \neq 0,

where

[| ω |]

represents the largest integer less than or equal to

| ω |

. Since

c

equals an odd number of half periods, the integral contributions over succeeding intervals will alternate in sign when the function

g

is positive and monotonically decreasing over

[a, \infty)

. The algorithm, described in Piessens et al. (1983), incorporates a global acceptance criterion (as defined by Malcolm and Simpson (1976)) together with the

ε

-algorithm (see Wynn (1956)) to perform extrapolation. The local error estimation is described by Piessens et al. (1983).

ω = 0

and

key = 1

, the routine uses the same algorithm as d01amf (with

epsrel = 0.0

In contrast to the other routines in Chapter D01, d01asf works only with an absolute error tolerance (epsabs). Over the interval

C_{k}

it attempts to satisfy the absolute accuracy requirement

{EPSA}_{k} = U_{k} \times epsabs,

where

U_{k} = (1 - p) p^{k - 1}

, for

k = 1, 2, \dots

and

p = 0.9

However, when difficulties occur during the integration over the

k

th sub-interval

C_{k}

such that the error flag

ierlst (k)

is nonzero, the accuracy requirement over subsequent intervals is relaxed. See Piessens et al. (1983) for more details.

4 References

Malcolm M A and Simpson R B (1976) Local versus global strategies for adaptive quadrature ACM Trans. Math. Software 1 129–146

Piessens R, de Doncker–Kapenga E, Überhuber C and Kahaner D (1983) QUADPACK, A Subroutine Package for Automatic Integration Springer–Verlag

Wynn P (1956) On a device for computing the

e_{m} (S_{n})

transformation Math. Tables Aids Comput. 10 91–96

5 Arguments

1: $g$ – real (Kind=nag_wp) Function, supplied by the user. External Procedure

g must return the value of the function

g

at a given point x.

The specification of g is:

Fortran Interface

Function g (

Real (Kind=nag_wp)	::	g
Real (Kind=nag_wp), Intent (In)	::	x

C Header Interface

double

g (const double *x)

1: $x$ – Real (Kind=nag_wp) Input: On entry: the point at which the function $g$ must be evaluated.

g must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d01asf is called. Arguments denoted as Input must not be changed by this procedure.

Note: g should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d01asf. If your code inadvertently does return any NaNs or infinities, d01asf is likely to produce unexpected results.

2: $a$ – Real (Kind=nag_wp) Input

On entry:

a

, the lower limit of integration.

3: $omega$ – Real (Kind=nag_wp) Input

On entry: the argument

ω

in the weight function of the transform.

4: $key$ – Integer Input

On entry: indicates which integral is to be computed.

$key = 1$: $w (x) = \cos (ω x)$ .
$key = 2$: $w (x) = \sin (ω x)$ .

Constraint:

key = 1

2

5: $epsabs$ – Real (Kind=nag_wp) Input

On entry: the absolute accuracy required. If epsabs is negative, the absolute value is used. See Section 7.

6: $result$ – Real (Kind=nag_wp) Output

On exit: the approximation to the integral

I

7: $abserr$ – Real (Kind=nag_wp) Output

On exit: an estimate of the modulus of the absolute error, which should be an upper bound for

| I - result |

8: $limlst$ – Integer Input

On entry: an upper bound on the number of intervals

C_{k}

needed for the integration.

Suggested value:

limlst = 50

is adequate for most problems.

Constraint:

limlst \geq 3

9: $lst$ – Integer Output

On exit: the number of intervals

C_{k}

actually used for the integration.

10: $erlst (limlst)$ – Real (Kind=nag_wp) array Output

On exit:

erlst (k)

contains the error estimate corresponding to the integral contribution over the interval

C_{k}

, for

k = 1, 2, \dots, lst

11: $rslst (limlst)$ – Real (Kind=nag_wp) array Output

On exit:

rslst (k)

contains the integral contribution over the interval

C_{k}

, for

k = 1, 2, \dots, lst

12: $ierlst (limlst)$ – Integer array Output

On exit:

ierlst (k)

contains the error flag corresponding to

rslst (k)

, for

k = 1, 2, \dots, lst

In the cases

ifail = 7

8

9

, additional information about the cause of the error can be obtained from the array ierlst, as follows:

$ierlst (k) = 1$: The maximum number of $subdivisions = \min (lw / 4, liw / 2)$ has been achieved on the $k$ th interval.
$ierlst (k) = 2$: Occurrence of round-off error is detected and prevents the tolerance imposed on the $k$ th interval from being achieved.
$ierlst (k) = 3$: Extremely bad integrand behaviour occurs at some points of the $k$ th interval.
$ierlst (k) = 4$: The integration procedure over the $k$ th interval does not converge (to within the required accuracy) due to round-off in the extrapolation procedure invoked on this interval. It is assumed that the result on this interval is the best which can be obtained.
$ierlst (k) = 5$: The integral over the $k$ th interval is probably divergent or slowly convergent. It must be noted that divergence can occur with any other value of $ierlst (k)$ .

13: $w (lw)$ – Real (Kind=nag_wp) array Workspace

14: $lw$ – Integer Input

On entry: the dimension of the array w as declared in the (sub)program from which d01asf is called. The value of lw (together with that of liw) imposes a bound on the number of sub-intervals into which each interval

C_{k}

may be divided by the routine. The number of sub-intervals cannot exceed

lw / 4

. The more difficult the integrand, the larger lw should be.

Suggested value: a value in the range

800

2000

is adequate for most problems.

Constraint:

lw \geq 4

15: $iw (liw)$ – Integer array Output

On exit:

iw (1)

contains the maximum number of sub-intervals actually used for integrating over any of the intervals

C_{k}

. The rest of the array is used as workspace.

16: $liw$ – Integer Input

On entry: the dimension of the array iw as declared in the (sub)program from which d01asf is called. The number of sub-intervals into which each interval

C_{k}

may be divided cannot exceed

liw / 2

Suggested value:

liw = lw / 2

Constraint:

liw \geq 2

17: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

−1

is recommended since useful values can be provided in some output arguments even when

ifail \neq 0

on exit. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

Note: in some cases d01asf may return useful information.

$ifail = 1$: The maximum number of subdivisions allowed with the given workspace has been reached without the accuracy requirements being achieved. Look at the integrand in order to determine the integration difficulties. If the position of a discontinuity or a singularity of algebraico-logarithmic type within the interval can be determined, the interval must be split up at this point and the integrator called on the subranges. If necessary, another integrator, which is designed for handling the type of difficulty involved, must be used. Alternatively, consider relaxing the accuracy requirements specified by epsabs or increasing the amount of workspace.

$ifail = 2$: Round-off error prevents the requested tolerance from being achieved: $epsabs = ⟨ value ⟩$ .

$ifail = 3$: Extremely bad integrand behaviour occurs around the sub-interval $(⟨ value ⟩, ⟨ value ⟩)$ . The same advice applies as in the case of $ifail = 1$ .

$ifail = 4$: Round-off error is detected in the extrapolation table.
The requested tolerance cannot be achieved because the extrapolation does not increase the accuracy satisfactorily; the returned result is the best that can be obtained. The same advice applies as in the case of $ifail = 1$ .
Look at the integrand in order to determine the integration difficulties. If the position of a local difficulty within the interval can be determined (e.g., a singularity of the integrand or its derivative, a peak, a discontinuity etc.) you will probably gain from splitting up the interval at this point and calling d01asf on the infinite subrange and an appropriate integrator on the finite subrange. Alternatively, consider relaxing the accuracy requirements specified by epsabs or increasing the amount of workspace.

$ifail = 5$: The integral is probably divergent or slowly convergent.

$ifail = 6$: On entry, $key = ⟨ value ⟩$ .
Constraint: $key \leq 2$ .

On entry, $key = ⟨ value ⟩$ .
Constraint: $key \geq 1$ .

On entry, $limlst = ⟨ value ⟩$ .
Constraint: $limlst \geq 3$ .

$ifail = 7$: Bad integration behaviour occurs within interval $K$ : $K = ⟨ value ⟩$ : $(⟨ value ⟩, ⟨ value ⟩)$ . Inspect ierlst for more details.

$ifail = 8$: The maximum number of intervals (limlst) has been reached: $limlst = ⟨ value ⟩$ . Inspect ierlst for more details.

$ifail = 9$: Extrapolation does not converge to the requested accuracy. Inspect ierlst for more details.

$ifail = 10$: On entry, $liw = ⟨ value ⟩$ .
Constraint: $liw \geq 2$ .

On entry, $lw = ⟨ value ⟩$ .
Constraint: $lw \geq 4$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.